Transcript Section 3B Putting Numbers in Perspective
Section 3C Dealing with Uncertainty
Pages 168-178
Motivating Story
(page 168) In 2001, government economists projected a cumulative surplus of $5.6 trillion in the US federal budget for the coming 10 years (through 2011)! That’s $20,000 for every man, woman and child in the US .
A mere two years later, the projected surplus had completely vanished.
What happened?
Assumptions included highly uncertain predictions about the future economy, future tax rates, and future spending. These uncertainties were diligently reported by the economists but not by the news media.
Understanding the nature of uncertainty will make you better equipped to assess the reliability of numbers in the news.
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Dealing with Uncertainty - Overview
Significant Digits Understanding Error Type – Random and Systematic Size – Absolute and Relative Accuracy and Precision Combining Measured Numbers
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Significant Digits –how we state measurements Suppose I measure my weight to be 133 pounds on a scale
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What is wrong with saying that I weigh 133.00 pounds?
5
133.00 incorrectly implies that I measured (and therefore know) my weight to the nearest one hundredth of a pound and I don’t!
The digits in a number that represent actual measurement and therefore have meaning are called significant digits.
When are digits significant?
Type of Digit
Nonzero digit (123.457)
Significance
Always significant Zeros that follow a nonzero digit and lie to the right of the decimal point (4.20 or 3.00) Always significant Zeros between nonzero digits (4002 or 3.06) or other significant zeros (first zero in 30.0
) Always significant Zeros to the left of the first nonzero digit ( 0.006
or 0.00052
) Never significant Zeros to the right of the last nonzero digit but before the decimal point (40,000 or 210) Not significant unless stated otherwise 3-C
Counting Significant Digits
Examples: 96.2 km/hr = 9.62
× 10 km/hr 3 significant digits (implies a measurement to the nearest .1 km/hr) 100.020 seconds = 1.00020 x 10 2 seconds 6 significant digits (implies a measurement to the nearest .001 sec.)
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Counting Significant Digits
Examples: 0.00098 mm =9.8
× 10 (-4) 2 significant digits (implies a measurement to the nearest .00001 mm)
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0.0002020 meter =2.020 x 10 (-4) 4 significant digits (implies a measurement to the nearest .0000001 m)
Counting Significant Digits
Examples: 300,000 =3 × 10 5 1 significant digit (implies a measurement to the nearest hundred thousand) 3.0000 x 10 5 = 300000 5 significant digits (implies a measurement to the nearest ten)
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Rounding with Significant Digits
Examples: 1452 x 9076.7; round to 2 significant digits = 13,179,368.4
with 2 significant digits : 13,000,000
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1452 x 9076.7; round to 4 significant digits = 13,179,368.4
with 4 significant digits: 13,180,000
Ever been to a math party?
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Understanding Error
•Errors can occur in many ways, but generally can be classified as one of two basic types: random or systematic errors.
•Whatever the source of an error, its size can be described in two different ways: as an absolute error , or as a relative error .
•Once a measurement is reported, we can evaluate it in terms of its accuracy and its precision.
Two Types of Measurement Error
3-C Random errors
occur because of random and inherently unpredictable events in the measurement process.
Systematic errors
the same amount.
occur when there is a problem in the measurement system that affects all measurements in the same way, such as making them all too low or too high by
Examples – Type of Error
Example: Weighing babies in a pediatricians office Shaking and crying baby introduces because a measurement could be “shaky” and easily misread.
random error A miscalibrated scale introduces because all measurements would be off by the same amount. (adjustable) systematic error
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Examples- Types of Error
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A count of SUVs passing through a busy intersection during a 20 minute period.
The average income of 25 people found by checking their tax returns.
Math parties are FUN!
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Size of Error – Absolute vs Relative
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A scale says Trig weighs 16.5 lbs but he really only weighs 15 lbs.
The same scale says my husbands weighs 185lbs, but he really weighs 183.5 lbs. Absolute Error in both cases is 1.5 lbs pounds Relative Error is 1.5/15 = .1 = 10% for Trig.
Relative Error Steve.
is 1.5/183.55 = .0082 = .82% for
Size of Error – Absolute vs Relative
3-C absolute error
= measured value – true value
relative error
absolute error true value measured value - true value true value
Absolute Error vs. Relative Error The government claims that a program costs $49.0 billion, but an audit shows that the true cost is $50.0 billion
3-C absolute error
= measured value – true value = $49.0 billion – $50.0 billion = $-1 billion
relative error
= absolute error true value = measured value true value true value -1 billion = 50 billion = .02 = 2%
Absolute Error vs. Relative Error Example: The label on a bag of dog food says “20 pounds,” but the true weight is only 18 pounds.
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absolute error = measured value – true value = 20 lbs – 18 lbs = 2 lbs
relative error
= absolute error true value 2 lbs 18lbs 100% 100% 11.1%
Accuracy vs. Precision
Accuracy
describes how closely a measurement approximates a true value. An accurate measurement is very close to the true value.
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describes the amount of detail in a measurement.
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Example Your true height is 62.50 inches . A tape measure that can be read to the nearest ⅛ inch gives your height as 62⅜ inches . A new laser device at the doctor’s office that gives reading to the nearest 0.05 inches gives your height as 62.90 inches .
Actual Height = 62.50 inches
Precision
Tape measure: Laser device: read to nearest 1/8” read to nearest .05” = 5/100” = 1/20” The laser device is more precise .
Accuracy
Tape measure: 62⅜ inches = 62.375” (absolute error = 62.375 – 62.50 = -.125” )
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Laser device: 62.90 inches (absolute error = 62.90 – 62.5 = .4” ) The tape measure is more accurate .
Math parties are REALLY FUN!
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Combining Measured Numbers
The population of your city is reported as 300,000 people. Your best friend moves to your city to share an apartment.
Is the new population 300,001?
NO!
300,001 = 300,000 + 1
Combining Measured Numbers
Rounding rule for addition or subtraction:
your answer to the same precision as the least precise number in the problem.
Round
Rounding rule for multiplication or division:
your answer to the same number of significant digits as the measurement with the fewest significant digits.
Round Note: You should do the rounding only after completing all the operations – NOT during the intermediate steps!!!
We round 300,001 to the same precision as 300,000.
So, we round to the hundred thousands to get 300,000.
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Combining Measured Numbers
A book written in 1962 states that the oldest Mayan ruins are 2000 years old. How old are they now (in 2007)?
The book is 2007-1962 = 45 years old.
We round to the nearest one year.
The ruins are 2000 + 45 = 2045 years old.
[2000 is the least precise (of 2000 and 45).] We round our answer to the nearest 1000 years.
The ruins are 2000 years old.
Combining Measured Numbers
The government in a city of 480,000 people plans to spend $112.4 million on a transportation project. Assuming all this money must come from taxes, what average amount must the city collect from each resident?
3-C $112,400,000 ÷ 480,000 people = $234.1666 per person 112.4 millions has 4 significant digits 480,000 has 2 significant digits So we round our answer to 2 significant digits.
$234.1666 rounds to $230 per person.